> > Anyone want a 46 PB? This would be a good way to get one.

>

to which Pauul said...

>

> Awesome! I'm thinking that 22, 31,

> 46, and 72 would be a good choice

> of four cardinalities to base my

> future instruments on. My

> present plan for the four

> fingerboards that come with the

> Rankin system is 22-tET, 31-tET,

> 22-out-of-46-tET, and 31-out-of-

> 72-tET.

>

Hi Paul,

Could you post the snail or email address for the Rankin

system again? This is sounding very interesting...

Regarding 46 which just showed up all over this list...

I got interested in 41 over the weekend, and decided that

when I got tired of meantone (31), that will be my trip

to the 9-limit.

What does 46 have that everyone seems gaga about? Is 46

better at 13 (recognizing that it doesn't do 11 or 13

uniquely, it does keep them off of each other, which

41 does not).

Is your 22-out-of-46 a sort of "nontempered" version of

the PB that you see in 22tet?

Is the 31-of-72 going to be Canasta?

Bob Valentine

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:

> What does 46 have that everyone seems gaga about?

>

> Is your 22-out-of-46 a sort of "nontempered" version of

> the PB that you see in 22tet?

Sort of . . . 22-out-of-46 is a "srutar", for Indian-type music. The

diaschisma 2048:2025 must vanish. It doesn't in 41.

> Is the 31-of-72 going to be Canasta?

Yes, with split frets, so that standard tuning can be used for the

open strings.

Bob Valentine wrote:

> What does 46 have that everyone seems gaga about? Is 46

> better at 13 (recognizing that it doesn't do 11 or 13

> uniquely, it does keep them off of each other, which

> 41 does not).

46 is almost 15-limit consistent, but tends to be overshadowed by its

neighbours 41 and 58. Well, 58 isn't casting much of a shadow right now,

but that's all going to change. So, every now and then we have to remind

ourselves that 46 is quite consonant as well, although those of us who

care about that are probably going back to 41 or 58 or 31 or 72.

Interestingly, fudging my temperament finding script so that 46 gets

included with the 15-limit ETs does bring 29+46 well into the rankings. I

wonder what else we could be missing by insisting on consistency.

Probably no killer temperaments, but maybe some keyboard mappings. For

example, a meantone-31 can do the 11-limit with 19 notes, but it doesn't

make the list because 31 is the only consistent temperament that works

with it. Also it isn't unique, but you can't have everything.

Graham

Graham wrote...

>46 is almost 15-limit consistent, but tends to be overshadowed by

>its neighbours 41 and 58. Well, 58 isn't casting much of a shadow

>right now, but that's all going to change.

What's going to change it?

-C.

Carl wrote:

> Graham wrote...

>

> >46 is almost 15-limit consistent, but tends to be overshadowed by

> >its neighbours 41 and 58. Well, 58 isn't casting much of a shadow

> >right now, but that's all going to change.

>

> What's going to change it?

Oh, it's bound to change. You can't keep a temperament like that down for

long. Wait until I get my hands on a ZTar ...

Graham

--- In tuning-math@y..., graham@m... wrote:

> Bob Valentine wrote:

>

> > What does 46 have that everyone seems gaga about? Is 46

> > better at 13

Yes, but I'm not planning on using it for 13-limit. More important is

that 46 is better than 41 in 5-limit (which is the main interval

flavor for Indian music). Though 34 is better still in the 5-limit,

46 allows me to access 7- and 11-flavors with much higher accuracy.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., graham@m... wrote:

> > Bob Valentine wrote:

> > > What does 46 have that everyone seems gaga about? Is 46

> > > better at 13

> Yes, but I'm not planning on using it for 13-limit. More important

is

> that 46 is better than 41 in 5-limit (which is the main interval

> flavor for Indian music). Though 34 is better still in the 5-limit,

> 46 allows me to access 7- and 11-flavors with much higher accuracy.

Here are relativized n-consistent goodness measures for odd n to 25,

for 41, 46 and 58:

41:

3 .67803586

5 1.380445520

7 .7433989824

9 .8004237371

11 .9169187332

13 1.010934526

15 1.010934526

17 1.138442704

19 1.042105199

21 1.042105199

23 1.399948567

25 1.399948567

46:

3 4.21934770

5 1.297511950

7 1.181094581

9 1.181094581

11 .8584628865

13 .8850299838

15 1.082290028

17 .9526171496

19 1.188323665

21 1.188323665

23 1.109794688

25 1.270499309

58:

3 4.18614652

5 2.499272068

7 1.270307523

9 1.270307523

11 .9740696116

13 .8435935409

15 .9018214272

17 .9309801788

19 1.393450457

21 1.393450457

23 1.295990287

25 1.721256452

We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits;

46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an

excellent system for 7, 9 or 11, and deserves some respect!

In-Reply-To: <9pbgq1+86qa@eGroups.com>

Gene wrote:

> Here are relativized n-consistent goodness measures for odd n to 25,

> for 41, 46 and 58:

How are these calculated? It looks like lower numbers are better, so it's

really a badness measure.

Graham

--- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <9pbgq1+86qa@e...>

> Gene wrote:

> > Here are relativized n-consistent goodness measures for odd n to

25,

> > for 41, 46 and 58:

> How are these calculated? It looks like lower numbers are better,

so it's

> really a badness measure.

This is the same w-consistent measure cons(w, n) introduced in

/tuning-math/message/860

And yes, it is a badness measure, but you could always take the

reciprocal. :)

In-Reply-To: <9pcpdn+pinr@eGroups.com>

> Gene wrote:

> This is the same w-consistent measure cons(w, n) introduced in

> /tuning-math/message/860

> And yes, it is a badness measure, but you could always take the

> reciprocal. :)

And "less than or equal to l" should be "less than or equal to w". I was

wondering how a prime could be less than 1.

It looks like h(q_i) is the number of steps in the ET for the ratio q_i.

So h(2) is the number of steps to the octave. So cons(w, n) assumes

consistency which -- warning! -- doesn't hold for 46-equal in the

15-limit, because there's one ambiguous interval.

This formula:

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))

has a parenthesis unclosed. From the one lower down, I think it should be

n^(1/d) * max(abs(n*log_2(q_i)) - h(q_i))

The n^(1/d) means you're scaling according to the size of the octave and

the number of prime dimensions.

Then you're taking the largest deviation for any interval within the

limit, right? In which case, the parenthesising should be

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i)))

and the other formula must be wrong.

That could be re-written

n^(1/d) * n * max(|tempered_pitch - just_pitch|)

where pitches are in octaves. Which can be simplified to

n^(1+1/d) * max(|tempered_pitch - just_pitch|)

Is that right? I think I almost understand it! So why the 1+1/d?

Graham

--- In tuning-math@y..., genewardsmith@j... wrote:

> 41 is clearly an

> excellent system for 7, 9 or 11, and deserves some respect!

Oh yes . . . but it's mentioned far more than 46 in the literature,

which is why I suspect Robert Valentine was puzzled about the 46

hoopla.

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > --- In tuning-math@y..., graham@m... wrote:

> > > Bob Valentine wrote:

>

> > > > What does 46 have that everyone seems gaga about? Is 46

> > > > better at 13

>

> > Yes, but I'm not planning on using it for 13-limit. More

important

> is

> > that 46 is better than 41 in 5-limit (which is the main interval

> > flavor for Indian music). Though 34 is better still in the 5-

limit,

> > 46 allows me to access 7- and 11-flavors with much higher

accuracy.

>

> Here are relativized n-consistent goodness measures for odd n to

25,

> for 41, 46 and 58:

>

> 41:

>

> 3 .67803586

> 5 1.380445520

> 7 .7433989824

> 9 .8004237371

> 11 .9169187332

> 13 1.010934526

> 15 1.010934526

> 17 1.138442704

> 19 1.042105199

> 21 1.042105199

> 23 1.399948567

> 25 1.399948567

>

> 46:

>

> 3 4.21934770

> 5 1.297511950

> 7 1.181094581

> 9 1.181094581

> 11 .8584628865

> 13 .8850299838

> 15 1.082290028

> 17 .9526171496

> 19 1.188323665

> 21 1.188323665

> 23 1.109794688

> 25 1.270499309

>

> 58:

>

> 3 4.18614652

> 5 2.499272068

> 7 1.270307523

> 9 1.270307523

> 11 .9740696116

> 13 .8435935409

> 15 .9018214272

> 17 .9309801788

> 19 1.393450457

> 21 1.393450457

> 23 1.295990287

> 25 1.721256452

>

> We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits;

> 46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an

> excellent system for 7, 9 or 11, and deserves some respect!

Gene, I don't know what you mean by n-consistent here. 46 is only

consistent through the 13-limit, so it's inconsistent in the 17-

limit. So how can 46 have a finite value for "17-consistent goodness"?

--- In tuning-math@y..., graham@m... wrote:

> And "less than or equal to l" should be "less than or equal to w".

I was

> wondering how a prime could be less than 1.

No, I mean that for those values of w, cons(w, n) is less than 1.

> It looks like h(q_i) is the number of steps in the ET for the ratio

q_i.

> So h(2) is the number of steps to the octave. So cons(w, n)

assumes

> consistency which -- warning! -- doesn't hold for 46-equal in the

> 15-limit, because there's one ambiguous interval.

Consistency always holds automatically the way I define things, since

I calculate everything based on the generators--that is, "h" is

defined not just by h(2) but also h(3), h(5) etc.

> The n^(1/d) means you're scaling according to the size of the

octave and

> the number of prime dimensions.

Right.

> That could be re-written

>

> n^(1/d) * n * max(|tempered_pitch - just_pitch|)

>

> where pitches are in octaves. Which can be simplified to

Right again.

> n^(1+1/d) * max(|tempered_pitch - just_pitch|)

>

> Is that right? I think I almost understand it! So why the 1+1/d?

If you are doing a simultaneous Diophantine approximation of d

different numbers, at least one of them irrational, by numbers with

denominator n, then we can find an infinite number of n such that all

of the numbers are within 1/n^(1+1/d) of the correct value. Choosing

a p-limit scale division involves the simultaneous approximation of d

values log_2(3), ..., log_2(p) by numbers of the form m/n. In this

case we add the condition that we also want to approximate

log_2(5/3), etc. but to get the dimensions right I still scale by

n^(1+1/d), since only d of these are independent.